Monotonicity of skew information and its applications in quantum resource theory

Abstract

We give an alternative proof of skew information via operator algebra approach and show its strong monotonicity under particular quantum TPCP maps. We then formulate a family of new resource measure if the resource can be characterized by a resource destroying map [1] and the free operation should be also modified. Our measure is easy-calculating and applicable to the coherence resource theory as well as quantum asymmetry theory. The operational interpretation needs to be further investigated.

Conditional quantum thermometry—enhancing precision by measuring less

“…Hence, the QFI of ρ β can be alternatively written as the covariance of the Hamiltonian H 12 ≡ H ⊗ 1 2 with respect to the conditional thermal separable state ρ β (3). By using the fact that Var ρ β {H 12 } = Var ρ β {H} and the asymmetry monotone of WYD skew information I α ( ρ β , H 12 ) ⩾ I α (ρ β , H) [35][36][37], we obtain (see appendix C for the proof. )…”

“…An asymmetry measure must satisfy two conditions: (i) it vanishes if and only if the state is symmetric, and (ii) it does not increase under all covariant operations. One quantity satisfying these conditions is the skew information [35][36][37][38][39][40].…”

“…To elucidate its properties further we then show that it can be understood as a non-passive state with useful resources [30][31][32][33][34]. To this end, we first relate its asymmetry, which is quantified by the Wigner-Yanase-Dyson (WYD) skew information [35][36][37][38][39][40], to its QFI. Then, focusing on a system undergoing unitary evolution, we discuss the state convertibility between the exact final state and the CTS constructed by the pointer states given by the evolved post-measurement state.…”

Quantifying Decoherence via Increases in Classicality

“…of ρ (with K i serving as a conserved observable) [88], which is now playing an increasingly interesting and important role in quantum theory [89][90][91][92][93][94][95][96][97]. In particular, the Wigner-Yanase skew information is monotone in the sense that [98,99] I(Φ(ρ), K) ≤ I(ρ, K)…”

“…It is remarkable that if is a Hermitian operator, then the summand in Equation ( 4 ) is precisely the celebrated Wigner–Yanase skew information of (with serving as a conserved observable) [ 88 ], which is now playing an increasingly interesting and important role in quantum theory [ 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 ]. In particular, the Wigner–Yanase skew information is monotone in the sense that [ 98 , 99 ] for any channel that does not disturb the observable K (i.e., ). This will be used to establish Proposition 1.…”